TL;DR: Try it first. If it's not enough, trash it, buy a cheap LGA 775 tower and upgrade it.
Your old tower
CS:GO is a pretty lightweight game, and if you really turn everything down, it can run on just about anything. This guide should allow you to squeeze a little out, and–if you're willing to go a bit farther (though there's been some debate as to if this is VAC safe) this guide should be able to help you get even a potato to run CS:GO.
Normally I would post the information you need (at least roughly) here, but there's simply too much content.
However, there's only so much you can do, and if you're still unsatified with your performance, I can really only recommend getting a different PC.
I'm really not a fan of trashing hardware–anyone who knows me or has read my posts knows I'm a massive fan of upgrading and used hardware. The problem is there's almost nothing that can be done with your PC.
To put it simply: it's too weak. Sure, it may be a quad core, and sure, it may boost up to 2.64 GHz, but with a 6.5W TDP, it is not a performer. CS:GO is heavily CPU dependent–especially single threaded performance–and this just can't keep up. It scores a mere 600 for a single thread in Passmark, getting thoroughly beaten by the 11-year-old Q6660, which scores a 924 without overclocking.
Problem 2: CPU upgrade
You can't. Though the spec page doesn't exactly make it clear, looking at some other J3710 equipped motherboards shows they are clearly soldered down.
Problem 3: GPU upgrade
HP prebuilt consumer towers are notorious for being annoying to install new GPUs into–however, even ignoring that, I can't recommend a GPU upgrade. There's almost zero information officially from HP as to if this is possible (I couldn't even find a picture of the motherboard), but we can makes some guesses. First, you're hard capped at 90W for your entire system, since as far as I can tell, the PSU is not upgradeable. This, combined with the fact that I wouldn't be surprised if your slot is capped at the lower 25W limit instead of the normal 75W limit as is fairly common among low end prebuilt, means I wouldn't trust anything above a GT 1030 to work. Though weak, the 1030 would still be enough for CS:GO. However, it costs $80–for which price, you can do much better.
A "new" rig
The key here is used components. We're going to start with a base platform. LGA 775 is old enough that they can be found really, really cheap, but is still good enough to provide an entry level experience. My personal choice is the Optiplex 755 with the Optiplex 780 in second, but any tower with support for a Core 2 Quad should work just fine. You should be able to find one for sale for $5-25; here's an eBay listing for $17.99. Your best bet to maximize value will be to search Craigslist and your local surplus markets (especially since you'll be able to save all that shipping cost); I find universities often have good stock of these older towers.
You'll need to check your platform first, but across the board, Core 2 Quads are very cheap. The Optiplex 755 can take a Q6600 (about $8.50), while the Optiplex 780 can take a Q8300 (about $7.00.) You can also try overclocking a hair to squeeze a little more performance using SetFSB.
Find the cheapest 4GB set of DDR2 available at the minimum speed required by your system. Make sure not to buy ECC or FBDIMM RAM, as these are not compatible. Craigslist may again be useful, or it may not. Here's a 2x2GB set for $8.10.
Buying cheap power supplies is never a great experience, but it'll have to do. Because there's so much variability, you'll need search around a bit. Check Craisglist and /r/buildapcsales. Here's a CX450W for $17.99.
Again, there's going to be a bit of shopping around here–what you can get will also depend on what PSU you got. The 9800GT for around $13 should do the trick, but I'd recommend the HD 5850 for around $25 if you can spare, or the HD 6950 for around $40 if you'd like to step it up a bit.
With a little time and effort, you've put together a decent little gaming PC for $50-100! I guarantee you this will play better than trying to upgrade your old PC for the same cost.